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https://github.com/langston-barrett/coq-big-o
A general yet easy-to-use formalization of Big O, Big Theta, and more based on seminormed vector spaces.
https://github.com/langston-barrett/coq-big-o
big-o complexity coq mathematics
Last synced: 3 months ago
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A general yet easy-to-use formalization of Big O, Big Theta, and more based on seminormed vector spaces.
- Host: GitHub
- URL: https://github.com/langston-barrett/coq-big-o
- Owner: langston-barrett
- License: mpl-2.0
- Created: 2017-02-16T06:24:59.000Z (almost 8 years ago)
- Default Branch: master
- Last Pushed: 2017-05-14T23:49:12.000Z (over 7 years ago)
- Last Synced: 2024-10-03T12:41:36.837Z (4 months ago)
- Topics: big-o, complexity, coq, mathematics
- Language: Coq
- Homepage:
- Size: 52.7 KB
- Stars: 34
- Watchers: 2
- Forks: 1
- Open Issues: 2
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Big O Notation for Coq
[](https://travis-ci.org/siddharthist/coq-big-o)
[](https://siddharthist.github.io/coq-big-o/html/toc.html)
A flexible yet easy-to-use formalization of Big O, Big Theta, and more
Bachmann-Landau notations based on seminormed vector spaces.
## Table of Contents
**Table of Contents**
- [Big O Notation for Coq](#big-o-notation-for-coq)
- [Table of Contents](#table-of-contents)
- [Features](#features)
- [Definitions](#definitions)
- [Theorems & Lemmas](#theorems--lemmas)
- [API Documentation](#api-documentation)
- [Installation](#installation)
- [Related Work](#related-work)
- [Contributing](#contributing)
## Features
### Definitions
- Bachmann-Landau notations
- Big O
- Big Ω
- Big Θ
- little o
- little ω
- (Optional) Unicode notation: `f ∈ ω(g) → g ∈ Ω(f)`
- Alternate definitions and proofs of their equivalence
### Theorems & Lemmas
This is not an exhaustive list:
- [x] Big Θ as an equivalence relation on functions
- [x] Big O as a partial ordering on functions
- [ ] Big Ω as a partial ordering on functions
- [x] Duality of Big O and Big Ω: f ∈ O(g) ↔ g ∈ Ω(f)
- [x] f ∈ o(g) → f ∈ O(g)
- [x] f ∈ ω(g) → f ∈ Ω(g)
- [x] Various facts about the compositions of functions
- [x] Indifference of these notations to multiplication by constants
- [ ] If you have ideas for useful lemmas, please open an issue!
## API Documentation
You can [view the documentation online][docs] or build it locally:
```
./configure && make html && firefox html/toc.html
```
to build the API documentation with `coqdoc`.
## Installation
You can build this package using the [Nix][nix] package manager:
```
nix-build . && ls result/lib/coq/8.5/user-contrib/BigO/
```
Alternatively, you can use the the standard
```
./configure && make
```
If you're using [Nix][nix], you can easily intergrate this library with your own
package's `default.nix` or `shell.nix`, and Coq should automatically find it.
```nix
{
stdenv,
coq,
pkgs ? import { }
}:
let
coq_big_o = with pkgs; callPackage (fetchFromGitHub {
owner = "siddharthist";
repo = "coq-big-o";
rev = "some commit hash"; # customize this
sha256 = "appropriate sha256 checksum"; # and this
}) { };
in stdenv.mkDerivation {
name = "my-coq-project";
src = ./.;
buildInputs = [ coq coq_big_o ];
...
}
```
Otherwise, just copy what you built to somewhere that Coq will find it.
### Related Work
I don't know of any. If anyone else is interested in formal complexity theory,
let me know!
This project leans heavily on the [math-classes][math-classes] library for
definitions of algebraic structures, specifically seminormed vector spaces.
## Contributing
Pull requests for fixes, new results, or anything else are welcome! Just run
```
nix-shell
```
to be dropped into a shell with all dependencies installed.
[nix]: https://nixos.org/nix/
[docs]: https://siddharthist.github.io/coq-big-o/html/toc.html
[math-classes]: https://github.com/math-classes/math-classes